1Quasi-homomorphisms

IV Bounded Cohomology

1.3 Poincare translation quasimorphism
We will later spend quite a lot of time studying actions on the circle. Thus, we
are naturally interested in the homeomorphism group of the sphere. We are
mostly interested in orientation-preserving actions only. Thus, we need to define
what it means for a homeomorphism ϕ: S
1
S
1
to be orientation-preserving.
The topologist will tell us that ϕ induces a map
ϕ
: H
1
(S
1
, Z) H
1
(S
1
, Z).
Since the homology group is generated by the fundamental class [
S
1
], invertibility
of
ϕ
implies
ϕ
([
S
1
]) =
±
[
S
1
]. Then we say
ϕ
is orientation-preserving if
ϕ
([S
1
]) = [S
1
].
However, this definition is practically useless if we want to do anything with
it. Instead, we can make use of the following definition:
Definition
(Positively-oriented triple)
.
We say a triple of points
x
1
, x
2
, x
3
S
1
is positively-oriented if they are distinct and ordered as follows:
x
1
x
2
x
3
More formally, recall that there is a natural covering map
π : R S
1
given by
quotienting by
Z
. Formally, we let
˜x
1
R
be any lift of
x
1
. Then let
˜x
2
, ˜x
3
be
the unique lifts of
x
2
and
x
3
respectively to [
˜x
1
, ˜x
1
+ 1). Then we say
x
1
, x
2
, x
3
are positively-oriented if ˜x
2
< ˜x
3
.
Definition
(Orientation-preserving map)
.
A map
S
1
S
1
is orientation-
preserving if it sends positively-oriented triples to positively-oriented triples.We
write
Homeo
+
(
S
1
) for the group of orientation-preserving homeomorphisms of
S
1
.
We can generate a large collection of homeomorphisms of
S
1
as follows for
any x R, we define the translation map
T
x
: R R
y 7→ y + x.
Identifying
S
1
with
R/Z
, we see that this gives a map
T
x
Homeo
+
(
S
1
). Of
course, if n is an integer, then T
x
= T
n+x
.
One can easily see that
Proposition.
Every lift
˜ϕ: R R
of an orientation preserving homeomorphism
ϕ: S
1
S
1
is a monotone increasing homeomorphism of
R
, commuting with
translation by Z, i.e.
˜ϕ T
m
= T
m
˜ϕ
for all m Z.
Conversely, any such map is a lift of an orientation-preserving homeomor-
phism.
We write
Homeo
+
Z
(
R
) for the set of all monotone increasing homeomorphisms
R R
that commute with
T
m
for all
m Z
. Then the above proposition says
there is a natural surjection
Homeo
+
Z
(
R
)
Homeo
+
(
S
1
). The kernel consists of
the translation-by-
m
maps for
m Z
. Thus,
Homeo
+
Z
(
R
) is a central extension
of Homeo
+
(S
1
). In other words, we have a short exact sequence
0 Z Homeo
+
Z
(R) Homeo
+
(S
1
) 0
i
p
.
The “central” part in the “central extension” refers to the fact that the image of
Z is in the center of Homeo
+
Z
(R).
Notation.
We write
Rot
for the group of rotations in
Homeo
+
(
S
1
). This
corresponds to the subgroup T
R
Homeo
+
Z
(R).
From a topological point of view, we can see that
Homeo
+
(
S
1
) retracts to
Rot
. More precisely, if we fix a basepoint
x
0
S
1
, and write
Homeo
+
(
S
1
, x
0
) for
the basepoint preserving maps, then every element in
Homeo
+
(
S
1
) is a product
of an element in
Rot
and one in
Homeo
+
(
S
1
, x
0
). Since
Homeo
+
(
S
1
, x
0
)
=
Homeo
+
([0, 1]) is contractible, it follows that Homeo
+
(S
1
) retracts to Rot.
A bit more fiddling around with the exact sequence above shows that
Homeo
+
Z
(
R
)
Homeo
+
(
S
1
) is in fact a universal covering space, and that
π
1
(Homeo
+
(S
1
)) = Z.
Lemma.
The function
F : Homeo
+
Z
(
R
)
R
given by
ϕ 7→ ϕ
(0) is a quasi-
homomorphism.
Proof. The commutation property of ϕ reads as follows:
ϕ(x + m) = ϕ(x) + m.
For a real number x R, we write
x = {x} + [x],
where 0 {x} < 1 and [x] = 1. Then we have
F (ϕ
1
ϕ
2
) = ϕ
1
(ϕ
2
(0))
= ϕ
1
(ϕ
2
(0))
= ϕ
1
({ϕ
2
(0)} + [ϕ
2
(0)])
= ϕ
1
({ϕ
2
(0)}) + [ϕ
2
(0)]
= ϕ
1
({ϕ
2
(0)}) + ϕ
2
(0) {ϕ
2
(0)}.
Since 0 {ϕ
2
(0)} < 1, we know that
ϕ
1
(0) ϕ
1
({ϕ
2
(0)}) < ϕ
1
(1) = ϕ
1
(0) + 1.
Then we have
ϕ
1
(0) + ϕ
2
(0) {ϕ
2
(0)} F (ϕ
1
ϕ
2
) < ϕ
1
(0) + 1 + ϕ
2
(0) {ϕ
2
(0)}.
So subtracting, we find that
1 −{ϕ
2
(0)} F (ϕ
1
ϕ
2
) F (ϕ
1
) F (ϕ
2
) < 1 {ϕ
2
(0)} 1.
So we find that
D(f) 1.
Definition
(Poincare translation quasimorphism)
.
The Poincare translation
quasimorphism T : Homeo
+
Z
(R) R is the homogenization of F .
It is easily seen that T (T
x
) = x. This allows us to define
Definition
(Rotation number)
.
The rotation number of
ϕ Homeo
+
(
S
1
) is
T ( ˜ϕ) mod Z R/Z, where ˜ϕ is a lift of ϕ to Homeo
+
Z
(R).
This rotation number contains a lot of interesting information about the
dynamics of the homeomorphism. For instance, minimal homeomorphisms of
S
1
are conjugate iff they have the same rotation number.
We will see that bounded cohomology allows us to generalize the rotation
number of a homeomorphism into an invariant for any group action.